There’s a subtly telling twist to the title of “The Who & The What,” the probing and funny but thematically sprawling new Ayad Akhtar play that aims to push more buttons than a teen on a texting bender.

The phrase comes from the postmodernist philosopher Jacques Derrida; the play’s central character, a fiery young Muslim-American woman named Zarina, invokes it to limn the difference between the descriptions affixed to a person (the “what”), and the genuine essence that lies behind such labels (the “who”).

And yet in Akhtar’s family saga, now getting its world premiere under Kimberly Senior’s direction at La Jolla Playhouse, the words also can represent something less high-minded: the voice of oppressive authority. This is who you are; this is what you will do.

That’s a reality faced not just by Zarina (a quietly forceful Monika Jolly) but by all four of the play’s characters in various ways. Akhtar — winner of the 2013 Pulitzer Prize in drama for “Disgraced” — has crafted a story that’s dense with intertwined desires, frustrations, expectations and resentments.

If it can feel at times as if the people in “The Who and the What” are modeled more to the story’s purposes than to the cause of believability, the piece is still a heady exploration of how one’s hoped-for path in life can crash against the ramparts of family and society. (Even when such barriers are partly unspoken and self-imposed.)

Akhtar’s story takes some cues from “The Taming of the Shrew,” Shakespeare’s story of romantic rebellion and submission — although there’s much less taming (and little tameness) to this play.

The 33-year-old Zarina is a frighteningly smart Harvard grad who’s working on a novel she describes vaguely as focusing on “gender politics.” What it’s really about, it develops, is more explosive: a tale of the Muslim prophet Muhammad that portrays him as subject to very human desires. (The book sounds akin to Nikos Kazantzakis’ incendiary “The Last Temptation of Christ,” made into a much-protested 1988 Scorsese movie.)

Zarina’s motivation is seemingly simple: “I hate what the faith has done to women,” she says. “For every story about (the prophet’s) generosity or his goodness, there’s another that’s used as an excuse to hide us. Erase us.”

But there’s more to the story: Zarina, like Kate in “Shrew,” has been subject to her father’s heartbreak-inducing marital meddling. Afzal (a magnetic Bernard White), the loving but tradition-minded Pakistani dad who has built a taxi-cab empire in Atlanta, already has quashed one of Zarina’s romances, and now is trying to arrange a new one — by posing (comically) as her on a Muslim dating service.

The man who turns up from Afzal’s trolling turns out to be quite a surprise catch: Eli (Kai Lennox), a white Muslim convert who runs a local mosque that doubles as a soup kitchen. He’s also a plumber who helps fix up houses for the needy. (Lennox turns in a winningly easygoing performance that’s probably as authentic as it can get for such an unlikely-seeming figure.)